On Product Difference Fibonacci Identities

Abstract

Simson's identity is a well-known Fibonacci identity in which the difference of certain order 2 products has a particularly pleasing form. Other old and beautiful identities of a similar nature are attributed to Catalan, Gelin and Cesàro, and Tagiuri. Catalan's identity can be described as a family of product difference Fibonacci identities of order 2 with 1 parameter. In Section 2 of this paper we present four families of product difference Fibonacci identities that involve higher order products. Being self-dual, each of these families may be regarded as a higher order analogue of Catalan's identity. We also state two conjectures that give the form of similar families of arbitrary order. In the final section we give other interesting product difference Fibonacci identities.

Simson Identity of Generalized m-step Fibonacci Numbers

“…A search of the literature turns up that there are many identities including Simson (Cassini), Catalan, d'Ocagne, Melham, Tagiuri, Gelin-Cesaro, Gould identities, see for example, [4,5,6,9,10,11,12,13].…”

Survey of Cubic Fibonacci Identities - When Cuboids Carry Weight

“…In spite of such long standing efforts, in the last decade further scrutiny of the Fibonacci sequence went unabated. Thus, Melham [68] introduced several generic identities and, among others, established:…”